Magnetic susceptibility computation

The spin-Hamiltonian concept, as proposed by Van Vleck [79], was introduced to EPR spectroscopy by Pryce [50, 74] and others [75, 80, 81]. H. H. Wickmann was the first to simulate paramagnetic Mossbauer spectra [82, 83], and E. Miinck and P. Debmnner published the first computer routine for magnetically split Mossbauer spectra [84] which then became the basis of other simulation packages [85]. Concise introductions to the related modem EPR techniques can be found in the book by Schweiger and Jeschke [86]. Magnetic susceptibility is covered in textbooks on molecular magnetism [87-89]. An introduction to MCD spectroscopy is provided by [90-92]. Various aspects of the analysis of applied-field Mossbauer spectra of paramagnetic systems have been covered by a number of articles and reviews in the past [93-100]. 

The general formula for GAmB(Z) has many pretty parts. To forestall any fear of using a complex expression, it is written here in Level 1 in a nearly exact version that can be easily taken apart for examination. This simplified version of the interaction formula holds for the case in which (1) relative differences in susceptibilities eA, eB, and em are small, (2) differences in magnetic susceptibilities are neglected, and (3) the velocities of light in media A and B are set equal to its velocity in medium m. These heuristic approximations will not be necessary, and in fact should be avoided in actual computation. 

A number of theoretical studies have been conducted to understand the mechanism of the Cope rearrangement.16 According to calculations by Houk and co-workers, the chairlike transition state is more stable than the boatlike transition state by 7.8 kcal/mol (Scheme l.XII). When Schleyer and colleagues performed calculations to compute the magnetic properties of the transition-state structures, transition states A and B had a magnetic susceptibility of—55.0 and—56.6, respectively. These values are comparable to that of benezene (—62.9), confirming the existence of an aromatic transition state in the Cope rearrangement. 

To confirm this finding based on the degree of the benzene ring C-C distortion, the magnetic susceptibility anisotropies and the relative energies for the benzene-fused five-membered heterocycles have been computed (Table 68). A similar approach was used by Schleyer et al. <1995AGE337> to evaluate the aromaticity of five-membered heterocycles using the ab initio methods. 

According to both the computed magnetic susceptibility anisotropies and relative energies, the benzo[. ]thiophenes lie between the benzo[ ]pyrroles and benzo[. ]furans the benzo[r]thiophene should be more stable than the benzo[r]furan benzo[ ]thiophene should be the more stable isomer. 

It can be concluded that the B3LYP/6-31G will produce high-quality structural parameters for the five-membered rings and their benzo derivatives. Based on the structural uniformity principle and magnetic susceptibility anisotropies, the predicted relative aromaticity of these systems is found to be reliable. From the computed values, the relative stability of thiophene, benzo[ ]thiophene, and benzo[f]thiophene is accurately predicted. 

The most convenient way to introduce the electric field is in terms of the vector potential A, by adding +e /c to the momentum operator in the Hamiltonian, with A chosen to give the field S = — (1/c) dA/dt = (ico/c)A. We shall use such an approach also when we compute the static magnetic susceptibility later, in Section... 

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